Question

Simplify - 1/root3+root2 - 2/root5-root3 - 3/root2- root5 Answer = 2root2

Answer 1

$\mathit{ \frac{1}{ \sqrt{3} + \sqrt{2} } - \frac{2}{ \sqrt{5} - \sqrt{3} } - \frac{3}{\sqrt{2} - \sqrt{5}} } \\ \\ \\ \mathit{\frac{1}{ \sqrt{3} + \sqrt{2} } * \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} - \frac{2}{ \sqrt{5} - \sqrt{3}} * \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} - \frac{3}{\sqrt{2} - \sqrt{5}} * \frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} + \sqrt{5}}} \\ \\ \\ \mathit{\sqrt{3} - \sqrt{2} - \frac{2*(\sqrt{5} + \sqrt{3})}{2} - \frac{3*(\sqrt{2} + \sqrt{5})}{-3}}$


$\mathit{\sqrt{3} - \sqrt{2} - \sqrt{5} - \sqrt{3} + \sqrt{2} + \sqrt{5}}\\ \\ \\ \mathit{0}$

Answer 2

Answer:

$\frac{-1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}=2\sqrt{2}$

Step-by-step explanation:

Given:

$\frac{-1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}$

We simplify by rationalizing the denominator,

$\frac{-1}{\sqrt{3}+\sqrt{2}}\times\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}\times\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}\times\frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}+\sqrt{5}}$

$\implies\frac{-1(\sqrt{3}-\sqrt{2})}{(\sqrt{3})^2-(\sqrt{2})^2}-\frac{2(\sqrt{5}+\sqrt{3})}{(\sqrt{5})^2-(\sqrt{3})^2}-\frac{3(\sqrt{2}+\sqrt{5})}{(\sqrt{2})^2-(\sqrt{5})^2}$

$\implies\frac{-\sqrt{3}+\sqrt{2}}{3-2}-\frac{2(\sqrt{5}+\sqrt{3})}{5-3}-\frac{3(\sqrt{2}+\sqrt{5})}{2-5}$

$\implies-\sqrt{3}+\sqrt{2}-\sqrt{5}+\sqrt{3}+\sqrt{2}+\sqrt{5}$

$\implies2\sqrt{2}$

Hence, $\frac{-1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}=2\sqrt{2}$